Post #2103465

2026-05-06 14:59 UTC

I have a category theory question that turned into a set theory question. If you have a functor F : Setᵒᵖ → Set which I promise is representable then you can easily get the the representing set, up to unique bijection, as F(1). What about for a functor F : Set → Set? If you do F(2) that tells you the powerset of the representing set. Is that enough to know the set? Is powerset injective on sets? If so, is it true in just ZF or do you need more? Is there a less horrendous way to get back the representing set of a promised-to-be-representable covariant functor, other than trying to invert the powerset operation?

Replies (6)

@lisyarus@mastodon.gamedev.place 2026-05-06 15:11

@julesh@mathstodon.xyz The powerset is a complete lattice wrt subset relation, and the largest element is the original set that the powerset was computed from. I'm not sure about the details but it sounds like that should be enough?

Open ##2278878
@ToucanIan@mathstodon.xyz 2026-05-06 15:19

@julesh@mathstodon.xyz P should be injective on sets by uniqueness of top elements in a sup lattice.

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@oantolin@mathstodon.xyz 2026-05-06 15:23

@julesh@mathstodon.xyz By the Yoneda lemma, if F is represented by some set A, then Nat(F, id) = id(A) = A. That's probably the easiest way to recover the representing object.

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@mudri@mathstodon.xyz 2026-05-06 15:47

@julesh@mathstodon.xyz ⋃𝒫X = X.

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@eigil@mathstodon.xyz 2026-05-06 21:02

@julesh@mathstodon.xyz X can be recovered from PX as the set of elements with exactly one other element below them (in the inclusion order, which can be recovered from the order on 2 and observing that Hom(X,-) preserves limits, so carries posets to posets).

Open ##2278885
@julian_kuelshammer@mathstodon.xyz 2026-05-07 17:03

@julesh@mathstodon.xyz To answer one of your questions, namely whether the power set function is injective (in the sense of cardinality): This depends on your axioms for set theory. Under the generalised continuum hypothesis it holds, without it, there are counterexamples, see https://mathoverflow.net/q/67473/15887

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